In summary, The Wall movie removed "Hey You" and "The Show Must Go On". An IMDB FAQ page discusses why here. It states:

A sequence was actually shot for "Hey You," but after a trial-viewing
of the film, Alan Parker and Roger Waters decided it was simply too
depressing and botched the film's pacing, so they had it removed. Some
of the footage used for "Hey You" was recycled into other parts of the
film, most noticeably into the "Another Brick in the Wall, Part 3"
sequence. You can view a low-fidelity version of the "Hey You"
sequence on the film's 2005 re-release DVD.

"The Show Must Go On" was left out for undisclosed reasons, although
one can infer it was also cut for the sake of pacing.

Three songs are included in the film, but not on the album:

When the Tigers Broke Free [New Song]

What Shall We Do Now? (an extended version of Empty Spaces)

5:11 AM (The Moment of Clarity) [Taken from The Pros and Cons of Hitchhiking, a Roger Waters concept album wrote at the same time as The Wall (although he released it later).]

Edit

Sticking with the links, and if anyone is interested, they can check out the awesome The Wall Analysis website. This website has an incredibly detailed breakdown of the meaning of every line of every song in both the movie and the album.

To give a snapshot of the detail this site contains, I'll take "When The Tigers Broke Free" as an example. Each song, including this one, has a "Song in a Sentence" description. For "When The Tigers Broke Free", it looks like this:

Pink recalls the last morning of his father's life, who was killed on
February 18, 1944 in the Battle of Anzio.

It then has seven or eight detailed paragraphs of explanation about the song, including what each line in the lyrics means, some trivia information, opinions from people on the site and an analysis of the song's context in the movie/album. Even posting this information for a single song would be too much (there is pages worth!), but an example of some of the information contained for "When The Tigers Broke Free" is:

Curiously enough, many Floydians rank "When the Tigers Broke Free"
among their favorite Wall songs...despite the fact that the song was
not part of the Wall canon until it appeared in the movie three years
after the album's release and subsequent concert tour. (On the DVD
commentary, Waters states that "Tigers" was written specifically for
the movie, though later he says that it was a song that was just lying
around. It's possible that it was a fragment during the album's
recording, and was finished and polished especially for the movie.)

For any Floyd fans who adore "The Wall", or anyone who just wants to get to understand either the album or movie better, it's a fantastic website to check out.

One of my all-time favorites! Song changes from album (taken from multiple sources, as well as my own notes):

"When the Tigers Broke Free"

New song, edited into two sections strictly for the film, but would later be released as one continuous song. The song was released as a single in 1982 and was later included on the 2001 compilation Echoes: The Best of Pink Floyd and on the 2004 re-release of The Final Cut.

"In the Flesh?"

Extended/re-mixed/lead vocal re-recorded by Geldof. The screaming kids entering the concert hall were added, which was inspired by Pink Floyd's run of Los Angeles, CA shows in 1975. Those shows were said to be the first inspiration for the concept of The Wall.

Re-mixed. Helicopter sounds dropped, teacher's lines re-recorded by Alex McAvoy. The poem young Pink was caught writing, which the teacher reads aloud, is "Money" from their Dark Side Of The Moon release.

"Another Brick in the Wall, Part 2"

Re-mixed with extra lead guitar, children's chorus part edited and shortened, teacher's lines re-recorded by McAvoy and interspersed within children's chorus portion.

"Mother"

Re-recorded completely with exception of guitar solo and its backing track. The lyric "Is it just a waste of time?" is replaced with "Mother, am I really dying?", which is what appeared on the original LP lyric sheet.

"What Shall We Do Now?"

A full-length song which begins with the music of, and a similar lyric to "Empty Spaces". This was intended to be on the original album, and in fact appears on the original LP lyric sheet. At the last minute, it was dropped in favour of the shorter "Empty Spaces" (which was originally intended as a reprise of "What Shall We Do Now"). A live version is on the album Is There Anybody Out There? The Wall Live 1980–81.

"Young Lust"

Screams added and phone call part removed. The phone call part was moved to the beginning of What Shall We Do Now

"One of My Turns"

Re-mixed. This song was inspired by the events on July 6, 1977 that directly led Roger Waters to write The Wall.

"Don't Leave Me Now"

Shortened and remixed.

"Another Brick in the Wall, Part 3"

Re-recorded completely with a slightly faster tempo.

"Goodbye Cruel World"

Unchanged.

"Is There Anybody Out There?"

Classical guitar re-recorded, this time played by David Gilmour with a leather pick, as opposed to the album version, which was played finger-style by an uncredited session guitarist.

"Nobody Home"

Musically unchanged, but with different clips from the TV set.

"Vera"

Unchanged.

"Bring the Boys Back Home"

Re-recorded completely with brass band and Welsh male vocal choir extended and without Waters' lead vocals.

"Comfortably Numb"

Re-mixed with screams added. Bass line partially different from album.

"In the Flesh"

Re-recorded completely with brass band and Geldof on lead vocals.

"Run Like Hell"

Re-mixed and shortened.

"Waiting for the Worms"

Shortened but with extended coda.

"5:11 AM (The Moment of Clarity)"

Geldof unaccompanied on lead vocals, sung in the bathroom stall just before The Trial sequence. The song is taken from The Pros and Cons of Hitch Hiking, a concept album Waters wrote simultaneously with The Wall, and later recorded solo. Geldof sings the lyrics to the melody of "Your Possible Pasts", a song intended for The Wall that later appeared on The Final Cut.

"Stop"

Re-recorded completely with Geldof unaccompanied on lead vocals. (The audio in the background of this scene is from Gary Yudman's introduction from The Wall Live at Earl's Court.)

"The Trial"

Re-mixed.

"Outside the Wall"

Re-recorded completely with brass band and Welsh male voice choir. Extended with a musical passage similar to "Southampton Dock" from The Final Cut.

The only songs from the album not used in the film are "Hey You" and "The Show Must Go On". "Hey You" was deleted as Waters and Parker felt the footage was too repetitive (eighty percent of the footage appears in montage sequences elsewhere) but available to view as in worn black and white work print form as a bonus feature on the DVD release under the name "Reel 13". It was also an "Extra" on the original Laserdisc release. It should be noted that the original Deluxe Letterbox Laserdisc release was the only release that featured a full-length commentary track by director Alan Parker. His was not included on any DVD/Blu-Ray releases to date, and no other Laserdisc versions of the film.

A soundtrack album from Columbia Records was listed in the film's end credits, but only a single containing "When the Tigers Broke Free" and the rerecorded "Bring the Boys Back Home" was released. "When the Tigers Broke Free" later became a bonus track on the band's 1983 album The Final Cut, an album Waters intended as an extension to The Wall. Guitarist David Gilmour, however, dismissed the album as a collection of songs that had been rejected for The Wall project, but were being recycled. The song, in the edit used for the single, also appears on the 2001 compilation album Echoes: The Best of Pink Floyd.

Additionally, look for the Immersion box set which includes loads of full-band demos, as well as some of the original demos Waters and Gilmour presented to the band. Some have completely different lyrics!